1. Field of the Invention
The present invention relates to a computer implemented method of generating a three dimensional hybrid grid of a heterogeneous formation crossed by one or more geometric discontinuities in order to carry out flow simulations.
The method applies more specifically to the formation of a grid of an underground reservoir crossed by one or more wells, or by fractures or faults, in order to model displacement of fluids such as hydrocarbons.
2. Description of the Prior Art
During development of a hydrocarbon reservoir, it is essential to be able to simulate gas or oil production profiles in order to assess cost-effectiveness, to validate or to optimize the position of the wells providing development and production. The repercussions of a technological or strategic change on the production of a reservoir also have to be estimated (location of the new wells to be drilled, optimization and selection during well completion, . . . ). Flow simulation calculations are therefore carried out within the reservoir which allow prediction of, according to the position of the wells and to certain petrophysical characteristics of the medium, such as porosity or permeability, the evolution with time of the proportions of water, gas and oil in the reservoir.
First of all, better comprehension of these physical phenomena requires 3D simulation of the multiphase flows in increasingly complex geologic structures, in the vicinity of several types of singularities such as stratifications, faults and complex wells. It is therefore essential to provide the numerical schemes with a correctly discretized field of study. Generation of a suitable grid then becomes a crucial element for oil reservoir simulators because it allows description of the geometry of the geologic structure studied by means of a representation in discrete elements. This complexity has to be taken into account by the grid which has to reproduce as accurately as possible the geology and all its heterogeneities.
Furthermore, to obtain a precise and realistic simulation, the grid has to be suited to the radial directions of flow in the vicinity of the wells, in the drainage zones.
Finally, grid modelling has made great advances during the past few years in other fields such as aeronautics, combustion in engines or structure mechanics. However, the gridding techniques used in these other fields cannot be applied as they are to the petroleum industry because the professional constraints are not the same. The numerical schemes are of finite difference type, which requires using a Cartesian grid, which is too simple to describe the complexity of the subsoil heterogeneities or, for most of them, of finite element type, suited to solve elliptic or parabolic problems, and not hyperbolic equations such as those obtained for the saturation. The finite difference and finite element type methods are therefore not suited for reservoir simulation, with only the finite volume type methods being suitable. The latter is the most commonly used method for reservoir simulation and modelling. It discretizes the field of study into control volumes on each one of which the unknown functions are approximated by constant functions. In the case of cell-centered finite volumes, the control volumes correspond to the cells and the discretization points are the centers of these cells. The advantage of this method is that the definition of the control volumes is readily generalized to any grid type, whether structured, unstructured or hybrid. Besides, the finite volume method remains close to the physics of the problem and meets the mass conservation principle (the mass balances of the various phases are written on each cell). Furthermore, it is particularly well-suited to the solution of hyperbolic type non-linear equations. It is therefore particularly suitable for solution of the hyperbolic saturation system. Therefore, hereafter cell-centered finite volume methods are used.
In fact, the grid allowing performing reservoir simulations has to be suited to:                describe the complexity of the geometry of the geologic structure studied,        the radial directions of flow in the vicinity of the wells, in the drainage zones, and        the simulations by means of cell-centered finite volume type methods.        
The grids which have been proposed and used to date in the petroleum industry are of three types: entirely structured, totally unstructured or hybrid, i.e. a mixture of these two grid types.
1. Structured grids are grids whose topology is fixed: each inner vertex is incident to a fixed number of cells and each cell is delimited by a fixed number of sides and edges. Cartesian grids (FIG. 1) can for example be mentioned, which are widely used in reservoir simulation, as well as CPG (Corner-Point-Geometry) type grids, described for example in French Patent 2,747,490 corresponding to U.S. Pat. No. 5,844,564 filed by the assignee, and grids of circular radial type (FIG. 2) allowing the drainage area of the wells to be modelled.
2. Unstructured grids have a completely arbitrary topology: a vertex of the grid can belong to any number of cells and each cell can have any number of edges or sides. The topological data therefore have to be permanently stored to explicitly know the neighboring nodes of each node. The memory cost involved by the use of an unstructured grid can therefore become very rapidly penalizing. However, they allow description of the geometry around the wells and representing complex geologic zones. The grids of PErpendicular BIssector (PEBI) or Voronoi type proposed in the following document can for example be mentioned:    Z. E. Heinemann, G. F. Heinemann and B. M. Tranta, “Modelling Heavily Faulted Reservoirs”, Proceedings of SPE Annual Technical Conferences, pp. 9-19, New Orleans, La., September 1998, SPE.
Structured grids have already shown their limits: their structured nature facilitates their use and implementation, but this also gives them a rigidity that does not allow all the geometric complexities of the geology to be represented. Unstructured grids are more flexible and they have allowed obtaining promising results but they still are 2.5D grids, that is the 3rd dimension is obtained only by vertical projection of the 2D result, and their lack of structure makes them more difficult to use.
3. To combine the advantages of the two approaches, structured and unstructured, while limiting the drawbacks thereof another type of grid has been considered: the hybrid grid. It is a combination of different grid types and it allows to make the most of their advantages, while trying to limit the drawbacks thereof.
A local refinement hybrid method has been proposed in:    O. A. Pedrosa and K. Aziz, “Use of Hybrid Grid in Reservoir Simulation”, Proceedings of SPE Middle East Oil Technical Conference, pp. 99-112, Bahrain, March 1985.
This method models a radial flow geometry around a well in a Cartesian type reservoir grid. The junction between the cells of the reservoir and of the well is then achieved using hexahedral type elements. However, the vertical trajectory followed by the center of the well must necessarily be located on a vertical line of vertices of the Cartesian reservoir grid.
To widen the field of application of this method, to take account of the vertical, horizontal wells and the faults in a Cartesian type reservoir grid, a new local refinement hybrid method has been proposed in:    S. Koeberber, “An Automatic Unstructured Control Volume Generation System for Geologically Complex Reservoirs”, Proceedings of the 14th SPE symposium on reservoir Simulation, pp. 241-252, Dallas, June 1997.
This method joins the reservoir grid and the well grid, or the reservoir grid blocks to the fault edges, by pyramidal, prismatic, hexahedral or tetrahedral type elements. However, the use of pyramidal or tetrahedral cells does not allow use of a cell-centered finite volume type method.
Finally, French Patents 2,802,324 and 2,801,710 filed by the assignee describe another type of hybrid model allowing accounting, in 2D and 2.5D, the complex geometry of the reservoirs and the radial directions of flow in the vicinity of the wells. This hybrid model allows very precise simulation of the radial nature of the flows in the vicinity of the wells by means of a cell-centered finite volume type method. It is structured nearly everywhere, which facilitates its use. The complexity inherent in the lack of structure is introduced only where it is strictly necessary, that is in the transition zones of reduced size. Calculations are fast and taking account of the directions of flow through the geometry of the wells increases their accuracy. Although this 2.5D hybrid grid has allowed taking a good step forward in reservoir simulation in complex geometries, the fact remains that this solution does not allow obtaining an entirely realistic simulation when the physical phenomena modelled are really 3D. It is the case, for example, for a local simulation around a well.
Furthermore, these hybrid grid construction techniques require creating a cavity between the reservoir grid and the well grid. S. Balaven-Clermidy describes, in “Génération de Maillages Hybrides Pour la Simulation des Réservoirs Pétroliers” (thesis, Ecole des Mines, Paris, December 2001), various methods for defining a cavity between the well grid and the reservoir grid: the minimum size cavity (by simple deactivation of the cells of the reservoir grid overlapping the well grid), the cavity obtained by expansion and the cavity referred to as Gabriel cavity. However, none of these methods is really satisfactory: the space created by the cavity does not allow the transition grid to keep an intermediate cell size between the well grid cells and the reservoir grid cells.